Final Results: Effects of Sex and Mixing on Male-Bias of TB

Description of Simulation Study

In this simulation study of pathogens spreading on contact networks, we sex-assortativity (r=0 to 0.6 by 0.1) and sex-trait ratios (\(\alpha\)) to see how the ratio of male-female cases changes. All simulation-parameter combinations show results from 250 replicates. Note: Sah network simulations finished for r=0, 0.6 by 0.1 but rewired networks currently running. Rewired results currently showing r=0, 0.6 by 0.1.

Assorted networks were generated in two ways: with methods in Sah et al. (2014) and with a re-wiring algorithm which selectively re-wires same-sex edges until the desired level of assortativity is reached (within a small value, \(\epsilon=0.05\)). All networks had mean degree of 10.

We investigated the following sex-traits: susceptibility (SUS), infectious period (IP), and transmissibility (TRA). We investigated different strengths of these sex-traits (\(\alpha\)).

Rates of susceptibility for nodes (SUS) was modeled as

Source -> Target	Overall transmission rate, \(\beta\)
F->F	\(\frac{2 \tau } {(\alpha + 1)}\)
M->M	\(\frac{2 \tau \alpha} {(\alpha + 1)}\)
M->F	\(\frac{2\tau } {(\alpha + 1)}\)
F->M	\(\frac{2 \tau \alpha} {(\alpha + 1)}\)

Rates of transmissibility (TRA) for nodes was modeled as

Source -> Target	Overall transmission rate, \(\beta\)
F->F	\(\frac{2 \tau } {(\alpha + 1)}\)
M->M	\(\frac{2 \tau \alpha} {(\alpha + 1)}\)
M->F	\(\frac{ 2 \tau \alpha} {(\alpha + 1)}\)
F->M	\(\frac{2 \tau} {(\alpha + 1)}\)

Infectious period (INF) for nodes modeled as

Source	Overall transmission rate, \(\beta\)
F	\(\frac{\gamma (\alpha + 1) } {2}\)
M	\(\frac{\gamma (\alpha + 1) } {2\alpha}\)

Each sex-trait was modeled separately.

Sensitivity parameters tested:

1. transmission rate (\(\tau=0.04, 0.075, 0.1\), \(R_0=1.5, 2.5, 3.5\))
2. model type (SIR, SLIR, SIRS, SLIRS). SIRS and SLIRS models ran for 200 time units.
3. Network type: SW (rewired), SF (rewired), Geometric (Sah Algorithm)

Response variables:

1. Male-bias: calculated differently for SIR/SLIR and SIRS/SLIRS. For models without recovery: number of male recovered nodes at end of simulation divided by number of female recovered nodes at end of simulation. For models with recovery: average ratio of male cases to female cases in last 100 time units for each simulation.
2. Epidemic duration: calculated for SIR/SLIR models as the number of time units before infectious population reached 0.
3. Total number infected: calculated for SIR/SLIR models as the total number of individuals that became infected before infectious population reached 0.
4. Prevalence of latent infection: calculated for SLIRS model as the size of the latent population at the last time step

Structure of Assorted Networks

Plots of networks generated with re-wiring algorithm. Node color represents different sexes (modules).

The sex assortativity coefficient is not reported in many studies. In the Kampala network study, we found sex assortativity coefficient to be 0.26. Horton et al. compared mixing by sex in 20 studies. They measured within-sex contact for adults and children. For adults, the IQR for men was .54 to .58 and for women was .57 to .63. The plot below shows the range .54 to .63 to represent what we feel the overall range could be for adults. For children, girls had sex assorted contact while boys did not probably showing the primarily female role of taking care of children.

We think these results, both the sex assortative coefficient from Kampala and Horton’s meta analysis, show that realistic levels of sex-assortativity in populations is from 0.2 to 0.4 and we will focus our attention on these parameter ranges.

We see from the plot below that, with the exception of the Sah algorithm, important network statistics change as assortativity increases. This is not surprising given that the Sah algorithm is designed to hold other parameters constant. This is something to keep in mind as other results are interpreted.

Changes in network statistics using the re-wiring algorithm to make networks of varying assortativity.

Model parameters: \(R_0\)

Below we examine the relationship between the epidemic threshold (\(R_0=1\)) as calculated in Kiss, Miller, & Simon (2017) and simulated epidemics on networks with different values of \(\tau\). Other parameters are given in figure caption. The epidemic threshold is given by:

\(\frac{\tau <K^2-K>}{\tau+\gamma <K>}>1\)

Comparison between analytically calculated \(R_0\) (Kiss, Miller, Simon 2017) and simulated epidemic size on non-assorted and assorted networks. Horizontal grey line shows where \(R_0=1\) and where ending epidemic size > 0. Vertical grey line approximately where \(R_0=1\). Results are consistent for SIR and SLIR models. No sex-trait heterogeneity is included in these simulated data. Other parameters: $\(\gamma=0.5; I_0=.01; \psi=0, 0.1; \delta=0.25, 1\cdot6\)

We see that the threshold matches up with simulated results (i.e., where \(R_0=1\)) epidemics are possible (i.e., outbreak size > \(I_0\)).

For \(\tau\) parameters used in this study, mean \(R_0\) on SAH networks for SLIR models is given below.

Tau values and estimated R0 for SLIR models simulated on SAH networks.

tau	mean_r0
0.040	1.504810
0.075	2.452056
0.100	3.374769

Model parameters: Equilibrium of latent infections and sex-bias

While the problem investigated with these models is inspired by TB, the models themselves are very simple, and not parameterized to any specific dataset. A marked trait of TB is that, globally, 25% of the world’s population has latent infection. Here, we investigate the equilibrium latent infections and case ratio for different infection rates and strengths of sex traits. We think these data points give an idea of the parameters that we should be focusing on with these models specifically for investigating male-bias in TB. The horizontal panels are faceted by assortativity level, and in between r=0.2, 0.3 is the level of sex-assortativity observed in a previous analysis of the urban social network studied by Dr Whalen and Dr Kiwanuka (and their teams) in Uganda.

Latent equilibrium and case-ratio on Sah networks.

Only Sah results are shown here but conclusions from simulations on rewired networks were similar.

It looks like \(\tau=0.075\) or \(\tau=0.1\) results in an equilibrium latent prevalence of 25%. In the next section, we look at results with \(\tau=0.075\) and then we consider the results with a slower spreading pathogen \(\tau=0.04\) and faster spreading pathogen \(\tau=0.1\).

Assortativity, Sex Traits, & Male Bias in SIR, SLIR, SIRS, and SLIRS models

Best guess for \(\tau\) (\(\tau=0.075, R_0=2.5\))

First, we show results for simulated male-bias with \(\tau=.075\) on SAH networks with varying \(\alpha\).

Results for the Sah networks above are representative of male-bias/assortativity results from other networks:

• Assortativity alone (where \(\alpha=1\) (no difference in sex trait)), even in very assorted networks, can not lead to male-bias in cases
• Higher male susceptibility and longer male infectious periods can lead to male bias without assortativity in some models but higher transmissibility can’t lead to male bias without assortativity
• Assortativity increases male bias when \(\alpha > 1\)
• In SIR and SLIR model, only higher male susceptibility can lead to male bias without assortativity
• Pathogens that do not have immunizing infections (SIRS, SLIRS) result in much more variable amounts of sex-bias in infections whereas SIR and SLIR result in lower variation in sex-bias
• Higher susceptibility seems to lead to more male-bias than other sex-traits, given a certain model type (e.g., for SLIR)

Results are similar on re-wired networks (only SLIR results shown below).

The graph above, which compares simulations on different network types, shows how higher male susceptibility is the only sex trait that can lead to male bias with assortativity.

With some assortativity, male bias increases with longer male infectious periods and higher male transmissibility but doesn’t seem to reach global levels of male-bias in cases.

How rate of transmission affects results

Here we show the sensitivity of these results to different \(\tau\) values for SLIR model.

alph_val	tau	r	SUS	TRA	IP
2	0.04	0.0	1.48	0.99	0.99
2	0.04	0.6	1.85	1.38	1.38
2	0.07	0.0	1.34	0.99	0.98
2	0.07	0.6	1.47	1.23	1.22
2	0.10	0.0	1.28	0.98	0.99
2	0.10	0.6	1.34	1.17	1.16

And here we show the sensitivity of these results to different \(\tau\) values for SLIRS model.

Faster spreading pathogens typically result in lower male-bias than slower spreading pathogens.

Parameter combinations leading to male-bias

The heat plots below show which parameter combinations lead to male:female case ratios greater than 1.25 given our best guess \(\tau\) estimate of 0.075.

The effects of sex-traits and assortativity on male-bias in simulations on SAH-networks. Results shown for low transmissability pathogen. Only ratios above 1.35 are shown to visualize which parameter combinations in which models can lead to male-bias. In WHO TB report, mean M:F case report ratio is 1.77 (+/- 1 SD 0.42, range 1.35, 2.19).

The effects of sex-traits and assortativity on male-bias in simulations on SAH-networks. Results shown for low transmissability pathogen. Only ratios above 1.35 are shown to visualize which parameter combinations in which models can lead to male-bias. In WHO TB report, mean M:F case report ratio is 1.77 (+/- 1 SD 0.42, range 1.35, 2.19).

The above plot shows possible combinations that lead to male-bias (1.7, 1.9). Any combinations without color lead to male:female case ratios < 1.1.

The effects of sex-traits and assortativity on male-bias in simulations on SAH-networks. Results shown for low transmissability pathogen. Only ratios above 1.35 are shown to visualize which parameter combinations in which models can lead to male-bias. In WHO TB report, mean M:F case report ratio is 1.77 (+/- 1 SD 0.42, range 1.35, 2.19).

Effects of network structures other than assortativity on male-bias (using SW networks)

What if sex-assortativity has the effect of reducing clustering, path length, and increasing degree-assortativity in the real-world? Here we use rewired small-world networks, which have much different structures compared with basic small-world networks, to see how these differences are related to male-bias.

Epidemic dynamics on assorted networks

SLIR

SIR

The mean peak, duration, and final size does not vary much (heat plots not shown) with assortativity.

The mean peak, duration, and final size does not vary much (heat plots not shown) with assortativity. To see the distribution of these values and results on different networks, see below box plots which are colored by assortativity values throughout this analysis.

The simulations on Sah networks suggest that assortativity does not variables much. But the other networks, which sometimes have increasing or decreasing clustering coefficients with increasing assortativity, sometimes indicate assortativity does affect final size.

Zooming in on SIR and SIRS specifically, the below plot shows how assortativity affects final size and equilibrium size specifically.

The plot above shows how different conclusions could be reached about the effects of assortativity depending on the characteristics of the network as assorativity is increased. If clustering increases as assortativity increases (SF), then increased assortativity could look to limit outbreak size. The opposite is true if clustering decreases as assortativity increases (SW).

Focusing on outbreaks on Sah networks, below we see whether differences in sex-traits impact epidemic dynamics.

In the plot above, SIR results are shown on Sah networks. They show increasing susceptibility in males relative to females can lower the overall fraction of the population that gets infected.

SLIRS

Finally, in SLIRS model, how does assortativity change equilibrium latent prevalence?

Increased male susceptibility relative to female susceptibility can decrease latent equilibrium infections.